Problem: Factor completely. $3 x^2 +30 x +75=$
First, we take a common factor of $3$. $3 x^2 +30 x +75=3(x^2 +10 x +25)$ Now, let's factor $x^2 +10 x +25$. Both $x^2$ and $25$ are perfect squares, since $x^2=({x})^2$ and $25=({5})^2$. Additionally, $10 x$ is twice the product of the roots of $x^2$ and $25$, since $10 x=2({x})({5})$. $x^2 +10 x +25 = ({x})^2 + 2({x})({5})+({5})^2$ So we can use the square of a sum pattern to factor: ${a}^2 + 2( a)( b)+ {b}^2 =({a} + {b})^2$ In this case, ${a}={x}$ and ${b}={5}$ : $ ({x})^2 + 2({x})({5})+({5})^2 =({x} +{5})^2$ $\begin{aligned} 3 x^2 +30 x +75 &=3(x^2 +10 x +25) \\\\ &=3(x +5)^2 \end{aligned}$ In conclusion, the complete factorization is $3(x +5)^2$ Remember that you can always check your factorization by expanding it.